In some of my previous work on mean values of Dirichlet L-functions, I came upon the following identity for the Gamma function:
\begin{equation}
\frac{\Gamma(a) \Gamma(1-a-b)}{\Gamma(1-b)}
+ \frac{\Gamma(b) \Gamma(1-a-b)}{ \Gamma(1-a)}
+ \frac{\Gamma(a) \Gamma(b)}{ \Gamma(a+b)} =
\pi^{\frac12}
\frac{\Gamma\left(\frac{ 1-a-b}{2}\right) }{\Gamma\left(\frac{a+b}{2}\right)}
\frac{\Gamma\left(\frac{a}{2}\right)}{\Gamma\left(\frac{1-a}{2}\right)}
\frac{\Gamma\left(\frac{b}{2}\right)}{\Gamma\left(\frac{1-b}{2}\right)}.
\end{equation}
As is often the case, once one knows such a formula should be true then it is easy to prove it. I give my proof below. My questions are 1) Has this formula been observed before? I have no idea how to search the literature for such a thing. 2) Is there a better proof? (Of course this is totally subjective, but one thing that would please me would be to avoid trigonometric functions since they do not appear in the formula.)

Note that the three ratios on the LHS are expressible as beta functions, e.g. $B(a,1-a-b)=\frac{\Gamma(a)\Gamma(1-a-b)}{\Gamma(1-b)}$
–
J. M.Sep 23 '10 at 5:31

I noticed that also, but I didn't see any beta function identities that seemed to help prove the formula. I'd love to see such an approach work though.
–
Matt YoungSep 23 '10 at 12:33

I should add, at one point I was convinced that there should be a natural proof of the identity using the usual integral representations of the beta functions on the left hand side and "simple" manipulations like changes of variable, but I didn't get that to work.
–
Matt YoungSep 23 '10 at 13:34

1

Maybe add the tag gamma-function? I have mixed feelings about proliferating such tags (this one applies to only 12 questions), but people select some of them to highlight their preferred subject areas.
–
Jim HumphreysSep 23 '10 at 14:46

One way to search the literature is to use the site functions.wolfram.com It is not an ideal site, as it has neither proofs nor citations. But if this identity is not there and is not easily derived from what is there, then you know that somebody who is paid to search the literature for such identities hasn't found it, too.
–
Kevin O'BryantSep 23 '10 at 22:51

Oh, and $A+B+C=\pi$ becomes $\alpha\beta\gamma=2i$. Now proving (1) is just a matter of multiplying out the right hand side and looking at the $8$ terms (two of them, namely $\alpha\beta\gamma$ and $\alpha^{-1}\beta^{-1}\gamma^{-1}$, cancel out, being $i$ and $-i$, respectively).

3) Here is how I would have done it 8 years ago: We can WLOG assume that $A$, $B$, $C$ are the angles of a triangle (this means that $A$, $B$, $C$ lie in the interval $\left[0,\pi\right]$, additionally to satisfying $A+B+C=\pi$), because everything is analytic (or by casebash). We denote the sides of this triangle by $a$, $b$, $c$ (so we forget about the old $a$, $b$, $c$), its semiperimeter $\dfrac{a+b+c}{2}$ by $s$, its area by $\Delta$ and its circumradius by $R$. Then, $\sin A=\dfrac{a}{2R}$ (by the Extended Law of Sines) and similarly $\sin B=\dfrac{b}{2R}$ and $\sin C=\dfrac{c}{2R}$, so that $\sin A+\sin B+\sin C=\dfrac{a}{2R}+\dfrac{b}{2R}+\dfrac{c}{2R}=\dfrac{a+b+c}{2R}=\dfrac{s}{R}$. On the other hand, one of the half-angle formulas shows that $\cos\dfrac{A}2=\sqrt{\dfrac{s\left(s-a\right)}{bc}}$, and similar formulas hold for $\cos\dfrac{B}2$ and $\cos\dfrac{C}2$, so that

A proof of the statement has already been given, so I will just add a small historical remark.
The left hand side of your identity is the Veneziano amplitude in the case of four identical scalar particles. The right hand side corresponds to another crossing symmetric four-point amplitude for scalar particles suggested by Virasoro in "Alternative Constructions of Crossing-Symmetric Amplitudes with Regge Behavior". These are known to coincide for the choice of parameters $a+b+c=1$ giving your identity. I would recommend the article "The birth of string theory" for a quick survey of the Veneziano model and its generalization to N scalar particles and related mathematics.

Jon Borwein recently told a class that the way to prove identities involving $\Gamma$ is to write the purported identity in the form $\Gamma(a)={\rm whatever}$ and then show that the right side satisfies the hypotheses of Bohr-Mollerup, that is, it's 1 when $a=1$, it satisfies $f(x+1)=xf(x)$, and it's log-convex. I have no idea whether that will work with this particular question, but maybe it's worth a try.

EDIT: It occurs to me that everything in the identity can be written in terms of Beta functions. Maybe it's easier to find if you look for it as a Beta-function identity.

By the way, is there an analogue of Bohr-Mollerup for the beta function?
–
Victor ProtsakSep 23 '10 at 7:14

I'll have to think about this approach to see how easy it is to employ. To my eye it has the disadvantage that it would not help in deriving such identities, but perhaps it is efficient at verifying them once they are known.
–
Matt YoungSep 23 '10 at 13:37

I decided to take the opposite approach to what Matt Y. asked for (sorry!), which was to see exactly how his proof depends on trig function identities. As it turns out, you can use trig identities to produce additional Gamma function identities like the original formula, though usually without being quite as clean and symmetric.

So I wondered how easy it would be to use known trig identities to produce additional formulas of this type, say with different denominators, and here is an example. Again using the reflection and multiplication formulas, you can show that
$$
\Gamma(s)\biggl( \frac{1}{2} + \cos\biggl(\frac{ 2\pi s}{3} \biggr) \biggr) = \pi \frac{3^{s-1/2} \Gamma\bigl(\frac{s}{3} \bigr)}{\Gamma \bigl( \frac{1-s}{3} \bigr) \Gamma \bigl( \frac{2-s}{3} \bigr)}.
$$
The trig identity I want to use (coming from the triple angle formulas for cosine) is
$$
2 \biggl( \frac{1}{2} + \cos\biggl( \frac{2x}{3} \biggr) \biggr)^3 - 3 \biggl( \frac{1}{2} + \cos \biggl( \frac{2x}{3} \biggr) \biggr)^2 + \sin^2(x) = 0.
$$
Take $x=\pi s$ and multiply by $\Gamma(s)^3$. Then after the dust settles this yields the identity
$$
2\pi 3^{3s-3/2}\frac{ \Gamma\bigl( \frac{s}{3} \bigr)^3}{ \Gamma\bigl( \frac{1-s}{3} \bigr)^3 \Gamma \bigl( \frac{2-s}{3} \bigr)^3} - 3^{2s} \frac{ \Gamma(s) \Gamma\bigl( \frac{s}{3} \bigr)^2}{\Gamma \bigl( \frac{1-s}{3} \bigr)^2 \Gamma \bigl( \frac{2-s}{3} \bigr)^2} + \frac{ \Gamma(s)}{\Gamma(1-s)^2} = 0.
$$
So yeah, it's not that pretty and we've only obtained a one-variable identity, but presumably one can find more examples of multivariable formulas in a similar fashion --- though you need to get the stars to align properly. What is nice about Matt Y.'s formula is that the powers of 2 that could be there (much like the powers of 3 here) nicely cancel away, leaving only $\Gamma$-values and a single power of $\pi$.

Very nice! I don't see offhand how to systematically get a "nice" gamma function identity out of a given trig identity though. Nevertheless, of course your method generates many identities.
–
Matt YoungSep 25 '10 at 1:49

I got to the same point and uttered the same sound: hmm... :)
–
Piero D'AnconaSep 23 '10 at 21:40

@Piero D'Ancona: The RHS seems annoying though. I thought maybe Legendre's doubling formula would make that ratio look nice, but no such luck. At the moment, this doesn't seem to be any more elegant than what's presented in the question.
–
Alex R.Sep 23 '10 at 21:41

I think there should be an additional $B( \frac{1-a-b}{2}, \frac{a}{2})$ factor in the numerator of the RHS.
–
Matt PapanikolasSep 24 '10 at 21:35

By adding/subtracting it with the original formula one gets an identity with the sum of two terms on each side. I didn't find it helpful but perhaps it sparks an idea with someone.

In the context of my original problem, the identity with + corresponds to a sum over even Dirichlet characters, while the identity in this answer corresponds to odd characters. The gamma factors on the right hand side are relevant for the functional equation of the Dirichlet L-functions which depends on the parity of the character.